Series

A series is simply the sum of the terms of a sequence. The following is a series of even numbers formed from the sequence 2, 4, 6, 8, . . . :

2 + 4 + 6 + 8 + . . .

A term of a series is identified by its position in the series. In the above series, 2 is the first term, 4 is the second term, etc. The ellipsis symbol (. . .) indicates that the series continues forever.

Example 1:

The sum of the squares of the first n positive integers 1 2 + 22 + 32 +Â…â€¦+ n2 is $\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ . What is the sum of the squares of the first 9 positive integers?

(A) 90

(B) 125

(C) 200

(D) 285

(E) 682

We are given a formula for the sum of the squares of the first n positive integers. Plugging n = 9 into this formula yields

$\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ = $\frac{9\left(9+1\right)\left(2Â·9+1\right)}{6}$ = $\frac{9\left(10\right)\left(19\right)}{6}$ = 285

Example 2:

For all integers x > 1, = 2x + (2x - 1) + (2x - 2)+Â…â€¦+2 + 1. What is the value of <3> Â· <2> ?

(A) 60

(B) 116

(C) 210

(D) 263

(E) 478

<3> = 2( 3) + (2 Â·3 - 1) + (2 Â·3 - 2) + (2 Â· 3 - 3) + (2 Â·3 - 4) + (2Â· 3 - 5) = 6 + 5+ 4 + 3+ 2 + 1 = 21

<2> = 2( 2) + (2Â· 2 - 1) + (2Â· 2 - 2) + (2Â· 2 - 3) = 4 + 3 + 2 + 1= 10

Hence, <3>Â·<2> = 21Â·10 = 210, and the answer is (C).